The Identification Problem for complex-valued Ornstein-Uhlenbeck Operators in $L^p(\mathbb{R}^d,\mathbb{C}^N)$

Abstract

In this paper we study perturbed Ornstein-Uhlenbeck operators \begin{align*}[\mathcal{L}_{\infty} v](x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2,\end{align*} for simultaneously diagonalizable matrices $A,B\in\mathbb{C}^{N,N}$. The unbounded drift term is defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain $\mathcal{D}(A_p)$ of the generator $A_p$ belonging to the Ornstein-Uhlenbeck semigroup coincides with the domain of $\mathcal{L}_{\infty}$ in $L^p(\mathbb{R}^d,\mathbb{C}^N)$ given by \begin{align*}\mathcal{D}^p_{\mathrm{loc}}(\mathcal{L}_0)=\left\{v\in W^{2,p}_{\mathrm{loc}}\cap L^p\mid A\triangle v+\langle S\cdot,\nabla v\rangle\in L^p\right\},\,1<p<\infty.\end{align*} One key assumption is a new $L^p$-antieigenvalue condition \begin{align*} \mu_1(A) > \frac{|p-2|}{p},\, 1<p<\infty, \,\mu_1(A) \text{ first antieigenvalue of $A$.}\end{align*} The proof utilizes the following ingredients. First we show the closedness of $\mathcal{L}_{\infty}$ in $L^p$ and derive $L^p$-resolvent estimates for $\mathcal{L}_{\infty}$. Then we prove that the Schwartz space is a core of $A_p$ and apply an $L^p$-solvability result of the resolvent equation for $A_p$. A second characterization shows that the maximal domain even coincides with \begin{align*}\mathcal{D}^p_{\mathrm{max}}(\mathcal{L}_0)=\{v\in W^{2,p}\mid \left\langle S\cdot,\nabla v\right\rangle\in L^p\},\,1<p<\infty.\end{align*} This second characterization is based on the first one, and its proof requires $L^p$-regularity for the Cauchy problem associated with $A_p$. Finally, we show a $W^{2,p}$-resolvent estimate for $\mathcal{L}_{\infty}$ and an $L^p$-estimate for the drift term $\langle S\cdot,\nabla v\rangle$.